Groundwater
Contamination
(Proceedings of the Symposium held during the Third IAHS
Scientific Assembly, Baltimore, MD, May 1989), IAHS Publ. no. 185, 1989
Application of the random walk method to simulate the
transport of kinetically adsorbing solutes
Albert J. Valocchi & Hernan A. M. Quinodoz
Department of Civil Engineering, University of Illinois, Urbana, Illinois 61801, USA
ABSTRACT
A new random walk method is presented for simulating
the transport of kinetically adsorbing solutes. The method
incorporates adsorption directly into the particle tracking
algorithm by utilizing an analytical formula for the probability
density function of the fraction of time a particle spends in the
aqueous phase. The new method is compared to two other techniques
for one-dimensional, nondispersive test cases. The new method is
shown to be more accurate and computationally efficient for moderate
to fast reaction rates. Future extension to two-dimensional,
randomly heterogeneous media is discussed.
INTRODUCTION
Adsorption reactions in porous media transport can be described
using either an equilibrium or kinetic approach; however, the former
has been most widely utilized to date due to its conceptual and
mathematical simplicity. Nonequilibrium adsorption phenomena have
been examined extensively on the laboratory scale for homogeneous
soil columns ; the important kinetic limitation at this scale
involves microscopic transport from the flowing pore fluid to the
soil-solution interface where the reaction occurs. Such kinetic
effects cause increased spreading and tailing of contaminant plumes
and breakthrough curves (Valocchi, 1985). However, the spatial
variability of real field aquifers causes additional kinetic
limitations involving macro-scale mixing among zones of differing
permeability. Data from a recent field study show deviations from
equilibrium sorption behavior even for low velocity, natural
gradient conditions (Roberts et al., 1986). Therefore, at present,
field-scale behavior of reactive contaminants is not wellunder stood.
Most theoretical studies of nonequilibrium transport are limited
to homogeneous media. However, preliminary analyses of perfectly
stratified aquifers show that there are strong interactions between
adsorption kinetics and hydrogeological variability; for example,
Valocchi (1988) has shown that deviations from local equilibrium
behavior diminish as the hydraulic conductivity heterogeneity
increases. We are currently extending these studies to two- and
three-dimensional, randomly heterogeneous aquifers. Our initial
goal is to develop a numerical transport model which will be used
to examine the relative contribution of kinetics and spatial
variability to the overall spreading of a solute plume. It is
crucial for the numerical technique to possess negligible numerical
dispersion and to be computationally efficient for three-dimensional
applications. With these two criteria in mind, we have selected the
random walk method. Also, Tompson et al. (1987) have recently
reported successful implementation of the random walk method for
35
36
A. J. Valocchi & H. A. M. Quinodoz
investigating macrodispersive phenomena for nonreactive transport
in three-dimensional, randomly heterogeneous aquifers.
Ahlstrom et al. (1977) and Kinzelbach (1988) have discussed
extension of random walk methods to handle chemically reactive
transport problems. However, for the simple first-order kinetic
adsorption reaction considered in our work, those techniques are
not always computationally efficient. The purpose of this paper
is to present a new random walk method for kinetically adsorbing
solute transport. The accuracy and efficiency of the new method
is demonstrated through several example simulations.
RANDOM WALK METHOD - THEORY
For simplicity, we will initially review the case of onedimensional, steady flow in a homogeneous porous medium. In this
case, the governing advection-dispersion equation (ADE) for a
nonreactive solute is
2
3c
_ 3 c
ôc
.. .
=
(1)
3Ï °x Tl ' V x Sx"
3x
where c is the aqueous concentration (aqueous species mass/fluid
volume), D x is the hydrodynamic dispersion coefficient (L / T ) , and
v x is the pore water velocity (L/T).
Next, consider a random walk where at each step the position of
an individual particle is adjusted according to
x (t + At) = x (t) + v At + Z 7 2D At
(2)
p
p
x
x
where x p (t) is the position of particle "p" at time t, At is the
time step, and Z is a random draw from a normally distributed random
variable having mean zero and standard deviation one. In the limit
of a large number of particles, the frequency distribution of the
number of particles per unit length satisfies the transport equation
(1). In practice, a finite number of particles are moved according
to the deterministic advective and stochastic dispersive steps
embodied in Eqn. (2). Then the concentration distribution is
estimated as
c(x,t) - M' • £ 7 7 ^
(3)
N(t)Ax
where M' is a normalizing constant, n(x,t) is the number of
particles at time t within a line segment Ax centered at x, and N(t)
is the total number of particles at time t. Tompson et al. (1987)
and Kinzelbach (1988) provide discussion of the formal equivalence
between Eqns. (1) and (2), as well as important generalizations to
three-dimensional heterogeneous media.
The governing ADE (1) must be modifed for the case where the
solute undergoes adsorption reactions. The modified ADE is
3c
_
+
3s
_ _
2
3 c
Dx
Vx
3c
_
3x
where s is the adsorbed species concentration (adsorbed species
(4)
Application of the random walk method
37
mass/fluid volume). Following Valocchi (1988) , we assume that the
adsorption rate (ds/dt) can be described by a first-order,
reversible, linear rate expression
|f = k f c - k r s
(5)
where k f (T _1 ) and k r (T"1) are the forward and reverse rate
coefficients, respectively.
Ahlstrom et al. (1977) presented a decoupled approach for
handling a broad class of chemical reactions in the random walk
method. Their decoupled approach is a continuum based one. In the
context of the single species adsorption problem here, particles
representing aqueous phase mass would advect and disperse according
to the nonreactive random walk [Eqn. (2)], then a smooth aqueous
phase concentration field would be calculated, and finally a
discretized form of Eqn. (5) would be solved locally in space for
the adsorbed phase concentration. The aqueous phase mass would then
need to be adjusted for any net adsorption or desorption before
proceeding to the next time step. Although the decoupled approach
is versatile, it is prone to numerical errors unless the time step
size is small. However, we now show how the linear rate expression
given by Eqn. (5) can be handled in a direct, particle-based fashion
by the random walk method.
Two-State Markov Chain
Using results presented by Parzen (1962), we can show that the
stochastic analogue of Eqn. (5) is a homogeneous, continuous - time,
two-state Markov chain. Consider a sequence of phase changes
experienced by a single particle. At any single instant, the
particle can exist in either of two states or phases: state 1
(aqueous phase) or state 2 (adsorbed phase). We denote this Markov
chain as (Y(t), t > 0} with state space {1,2}. The rates at which
the particle leaves states 1 and 2 (so-called intensities of
passage) are kf and k r , respectively; the time the particle spends
in either state prior to making a transition is an exponentially
distributed random variable with a parameter equal to the
appropriate rate of change (kf or k r ) .
A continuous-time Markov chain is completely specified if we know
both the intensities of passage and the transition probability
functions. These functions measure the probability of transition
between any combination of the two states, given the initial and
final times. There are four such combinations, namely 1-1, 1-2, 21, 2-2. When the Markov chain is homogeneous, it is only the
difference t between the initial and final times that is important.
The transition probability functions are usually found by solving a
system of differential equations, called the Kolmogorov differential
equations, which for our problem read:
dP lfl (t)/dt = -kfP]L x (t) + k r p l j 2 (t)
d p l f 2 ( t ) / d t - -k r p 1 > 2 (t) + k f P l , l ( t )
(6)
dp 2 2(t)/dt = -k r p 2 2 (t) + k f p 2 x (t)
dp
(t)/dt- -k f P
(t) + \ p 2 > 2 ( t )
38
A. J. Valocchi & H. A. M. Quinodoz
The notation used for the above probabilities uses as sub-indices
the two phases associated with the transition: the first index is
for the initial phase and the second index for the final phase.
Note that each of the above equations can be interpreted as a
stochastic gain-loss equation for the transition probabilities
between two states. Parzen (1962) solved the above system of
equations and obtained the four transition probabilities; his
results are as follows :
p 2 x (t) = P[Y(t+s) - 1 J Y(s) - 2]
„(t) - P[Y(t+s) = 2 | Y(s) - 1]
1,2
Pl
P
l,l
=
P
2,2 -
l
' Pl,2
X
" P 2,l
k_ + k
f
r
k_ + k
f
r
1 -e
(k^+k )tf r
-(k_+k )tf r
1 -e
(7)
(8)
(9)
(10)
The above four transition probabilities do not depend upon the
number of phase changes during time t; that number can take on any
value compatible with the initial and final state. However, because
the waiting time in each state is exponentially distributed, we can
easily compute the probability of zero phase changes within time t.
Denoting this by a superscript z, we have
-k t
r
-kft
(11)
2,2
1,1
Finally, we develop the correspondence between Eqn. (5) and the
two-state, continuous-time Markov chain described above. Let o^(0)
and «2(0) denote the probability that a particle is in state 1 and
2, respectively, at time equal to zero. Then the probability that a
particle is in the adsorbed phase (state 2) at time t, P2(t), is
P 2 (t) = P;L 2 (t)
0^(0) + p 2
2 (t)
a 2 (0)
A similar expression can be derived for P]_(t). Differentiating the
above equation with respect to time, and substituting (6) for
dp.. „/dt and dp„ ,-,/dt yields
Tz
P
2 ( t > " k f Pl<fc> " k r P 2 ^ >
(12)
Equation (12) is identical to Eqn. (5) if the probability that a
particle is in a particular phase is interpreted as the phase
concentration. In practice, the time history of phase changes of a
large number of independent particles can be simulated; P]_(t) is
then estimated as the fraction of particles in phase 1 at time t.
Application of the random walk method
39
RANDOM WALK METHOD-IMPLEMENTATION
In this section, we present three alternative approaches to
implementing the two-state, continuous-time, Markov chain
representation of the adsorption-desorption reaction. Depending
upon the nonuniformity of the velocity field, a certain maximum
timestep (At) will be dictated for the advection and dispersive
steps in Eqn. (2) (Tompson et al., 1987). Since only aqueous-phase
particles are mobile, the key problem is to determine the fraction
of the timestep that each particle remains in the aqueous phase.
Method 1:
Continuous Time History Method
This method takes the natural approach of simulating the history of
phase changes for each particle during At. This can be accomplished
in a straightforward manner by generating a sequence of exponential
waiting times with parameters alternating between kf and k r (or vice
versa if the initial state of the particle is adsorbed). The
process stops when the total time elapsed is larger than At; at that
point the waiting times for each state are summed and the particle
is advected/dispersed using a time step equal to the time spent in
the aqueous phase.
The method is computationally simple since the exponential
distribution has a closed form inverse which allows efficient
generation of random variables by the inverse transform method
(Rubinstein, 1981). The main difficulty occurs for fast reaction
rates, which requires generation of an unacceptably large number of
waiting times.
Method 2:
Small Time Step Method
In the limit of small (kfAt + k r At), the transition probabilities,
Eqns. (7) and (8), become
p
- k f At
P
2 1=
k
r
At
(13)
For a given kf and k r , Eqn. (13) holds for "small enough" At.
Since the number of phase transitions can be considered a Poisson
process, the probability of having more than one phase change in a
small At is negligible (Parzen, 1962). Therefore, a particle is
assumed to remain in its initial state for the entire At, and its
final state is adjusted by a Bernoulli trial where a uniform (0,1)
random variate is compared to the appropriate transition probability
in Eqn. (13). This method has been previously presented by
Kinzelbach (1988) . The main drawback of this approach is that At
must be chosen to guarantee that p^ 2 anc* P2 1 above are much
smaller than one; for fast reactions, this may require an
excessively small At.
Method 3:
Arbitrary Time Step Method
For an arbitrary At, Keller and Giddings (1960) have determined
probability distributions for /3(At), the fraction of time that a
particle spends in the aqueous phase. Their results for the
probability density functions depend upon the initial and final
phases and are as follows :
f
l,l(/3)
fx
2(/9)
=
A. J. Valocchi &H.A. M. Quinodoz
1/2
[Sfl
exp[-a(l-0) " b ^] I1(74ab^(l-(S))
- b exp[-a(l-/3) - b/3] I o (,/4ab0(l-0))
f
2,l(/9)
=
bLfl,2(^)
f
2 , 2 ( ^ " jt
f
l,l(/,)
(14)
(15)
( 1 6 )
(17)
where a = kfAt and b = k r At, and I Q and X-y are modified Bessel
functions of order 0 and 1, respectively.
The algorithm involves three different steps. First, we
determine if there is a phase change by performing a Bernoulli trial
using the appropriate zero-transition probabilities in Eqn. (11).
Second, if there is a phase change, we determine the final phase by
performing a Bernoulli trial using the transition probabilities in
Eqns. (7)-(10), which are first adjusted to exclude the probability
of zero-transition used in the first step. Finally, we determine
the time spent in the aqueous phase by applying the inverse
transform method to the cumulative distribution functions derived
from the density functions in Eqns. (14)-(17). These three steps
will result in a value of /3 between zero and one for each particle.
The advective and dispersive steps can then be performed for each
particle with /3At instead of At in Eqn. (2). A distinct advantage
of this method over the previous two techniques is that the overall
computational effort is more or less independent of the reaction
rate.
RESULTS
The three methods described in the previous section were tested for
two different cases of one-dimensional, steady, uniform flow without
dispersion. The first case considered a constant input of solute at
the upstream boundary. This case was chosen because of its
simplicity and because of the availability of an analytical solution
(Bolt, 1982). Simulations with all three methods used 10,000
particles, a pore water velocity v=l, and kf = k r = 1.0.
Concentration profiles were obtained for selected times and compared
to the analytical solution. A graphical comparison of the profiles
at T - 100 is shown in Fig. 1 for Method 3. The fluctuating nature
of the numerical solution represents statistical noise associated
with the finite number of particles used (Kinzelbach, 1988). The
other two methods produced results of nearly identical quality,
although the required CPU time was about 5 times larger for Method
2, for which it was necessary to reduce the time step by a factor of
10 relative to the other two methods.
The second case consists of an instantaneous point discharge of
aqueous-phase solute as the initial condition. This case was
selected because there is an analytical solution available for the
evolution in time of the first two spatial moments of the cloud of
particles in the mobile phase (Valocchi, 1988). The test used
different rates for each phase, while keeping the ratio kf/k r equal
to 0.5. The individual rates were varied to cover the range of slow
to fast reaction rates. The flow velocity was v=l, and the number
Application of the random walk method
41
C/Co
Fig. 1 Comparison between random walk (Method 3) and analytical
solutions at T=100.
of particles was varied in the different experiments.
All three methods are compared using the first two moments of the
position of the particle cloud: mean and variance, as well as the
total CPU time on a DEC-Vaxstation II. The error in the values of
the mean position were very small in general, and only results for
the variance are discussed here. Table 1 summarizes the results for
a time T=100. Results of comparable quality were obtained with both
the continuous time history (Method 1) and arbitrary time step
(Method 3) methods, while the small time step method (Method 2)
produced larger errors. Computing times are comparable only for
slow reaction rates: for moderate and fast rates Method 2 is the
most time consuming, followed by Method 1. The accuracy of Method 2
is improved when At is reduced, but this results in an increased CPU
time. Overall, the most efficient and accurate method for all
reaction rates is Method 3.
TABLE 1
Relative Error in Variance of Particle Positions at T=100
Time
Relative
CPU time
Method
step
error (%)
(min)
Moderate reaction rate
kr = 1.0
N = 100,000
Fast reaction rate
kr = 10.
N = 10,000
1
2
3
1.
1.
1.
0.36
5.92
0.79
141.
121.
127.
1
2
3
1.
0.1
1.
0.34
7.10
0.55
169.
1183.
167.
1
2
3
i-i
Slow reaction rate
kr = 0.1
N = 100,000
1.42
8.32
1.49
45.
889.
22.
0.01
1.
The agreement between numerical and analytical solutions is very
good; however, the relative error in the variance estimation is
sensitive to the number of particles (N) and to the reaction rates.
42
A. J. Valocchi & H. A. M. Quinodoz
Table 1 shows the relative error is larger than 1% for all methods
when using 10,000 particles. A single run with fast rates was
repeated with 100,000 particles using Method 3 which effectively
lowered the error to 0.37%.
FUTURE DIRECTIONS
Because of its accuracy, robustness, and efficiency, Method 3 has
been used in the development of a two-dimensional transport code.
The code has been modified and optimized to run on an Alliant FX-8
computer which is a shared-memory, parallel (8 processors) machine.
About 95% of the code is multi-taskable and sample test problems run
approximately forty times faster on the Alliant than on the
Vaxstation. We next plan to use the code to examine the impact of
various rate coefficients upon the particle cloud variance for fixed
patterns of heterogeneity. We will initially use spatially uniform
rate parameters. Although the methods described above are valid
for spatially variable reaction rates, computational implementation
would be considerably more difficult.
ACKNOWLEDGMENTS
This paper is based upon work supported by the
U.S. Geological Survey under Grant No. INT 14-08-G1299 and by the
National Center for Supercomputing Applications.
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